Mathematics > Optimization and Control

Title:Symmetry-free SDP Relaxations for Affine Subspace Clustering

Abstract: We consider clustering problems where the goal is to determine an optimal
partition of a given point set in Euclidean space in terms of a collection of
affine subspaces. While there is vast literature on heuristics for this kind of
problem, such approaches are known to be susceptible to poor initializations
and getting trapped in bad local optima. We alleviate these issues by
introducing a semidefinite relaxation based on Lasserre's method of moments.
While a similiar approach is known for classical Euclidean clustering problems,
a generalization to our more general subspace scenario is not straightforward,
due to the high symmetry of the objective function that weakens any convex
relaxation. We therefore introduce a new mechanism for symmetry breaking based
on covering the feasible region with polytopes. Additionally, we introduce and
analyze a deterministic rounding heuristic.